Pascal's Triangle and Combinatorics

Pascal's Triangle can be used to easily work out the number of permutations for a given number of "ingredients" and "places". For example the above diagram highlights that the number of permutations for 3 ingredients over 3 places equals 27:

AAA

AAB

ABA

BAA

AAC

ACA

CAA

ABC

ACB

BBB

BBA

BAB

ABB

BBC

BCB

CBB

BCA

BAC

CCC

CCA

CAC

ACC

CCB

CCB

BCC

CAB

CBA

One simply follows the diagonals down from each side. The circle where they meet shows the answer. The smaller number in the circle indicates the number of different Ratios possible for the ingredients and places. In this case it is 10:

ABC

AAA

BBB

CCC

AAB

BBA

CCA

AAC

BBC

CCB

Note that the small black "Ratio" numbers are identical with the numbers of Pascal's Triangle.

The larger blue "Permutation" numbers follow a simple pattern. The diagonal line from a particular Ingredient Number contains all the powers of that number from 1 to as far as the triangle is extended. For example the 2 Ingredient Line contains all the powers of 2 ie: 1,2,4,8,16,32...etc and the 3 Ingredient Line contains all the powers of 3 ie: 1, 3, 9, 27, 81...etc.

The diagnal line from a particular Place Number contains all the numbers of a particular power from 1 to as far as the triangle is extended. For example the 2 Place Line includes all the Square Numbers ie: 1, 4, 9, 16, 25...etc and the 3 Place Line contains all the cube numbers ie: 1, 8, 27, 64...etc.

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full descriptions and detailed maps of each of the trekking routes - including altitudes and GPS waypoint readings - by somebody who has climbed them all!
Is the only book with an extensive and unbiased review of the trekking agencies, both in Tanzania (Arusha, Moshi and Marangu) and those tour operators in the UK/US/Europe/Australia/ New Zealand/South Africa.
Is the only book with town and city guides to Arusha, Moshi and Marangu, the three towns and villages that will be your base before and after your climb
Has the most comprehensive guide to preparing for your trek, from what to take, what to wear, how to book your trek and flights, and how much it's all going to cost; everything, in fact, to help you get ready for the biggest adventure of your life!
Provides the most detailed background information on the mountain, including the history, geography, geology and topography of Kilimanjaro, as well as a study of the Chagga tribe that live at the foot of the mountain.
Full run-down of Kilimanjaro's nature including a full-colour flora guide.
A full description with maps of the Mount Meru trek.

We also provide:

A thorough health and fitness section to help you stay healthy in East Africa, increase your chances of making it to the top and ensure you make it back down again.
In-depth study of altitude sickness, or AMS - the number one reason why people fail to make it to the summit - including causes and prevention;
City guides to Nairobi and Dar es Salaam, the two cities that you are most likely to fly into, as well as an introduction to Kilimanjaro International Airport;
A thorough practical information guide to travelling around Tanzania, including a look at visa requirements, currency, budgeting, transport, food etc);
Advice on how to look after your porters and crew on the trek;
A look at the culture of the local Chagga people who inhabit Kilimanjaro's lower slopes;
Plenty of photos and illustrations from both contemporary and historical sources, including some wonderful pen-and-ink sketches by the first Europeans to see the mountain, way back in the nineteenth century.
Over thirty maps covering every stage of climbing and descending from the mountain.

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This guide to treks and excursions around Kilimanjaro provides information on getting to Kilimanjaro from Europe, North America and Australasia, trekking preparations, where to stay and eat along the trails, employing a guide or porter, the environment and health and safety.Click Here to Learn More

A folded, illustrated map of Arusha National Parkin Tanzania, showing tourist information, roads, and illustrations of various trees and plants that can be seen. Also includes a detailed day to day trip itinerary for climbing Mt. Meru, along with information on regulations, mountain routes, and a map of the trails around Mt. Meru. Legend and text is in English.Click Here to Learn More

This is a book about Sikh religion's founder Guru Nanak and his main spiritual composition called Japji. The book contains a brief life sketch of Guru Nanak, his basic beliefs, translation of Japji text, and passage by passage commentary. The book also contains a 14 point model for leading spiritual life in accordance with Guru Nanak's teachings. The book is unique in the sense that it contains the first gender-neutral presentation of Japji in modern North American English.Click Here to Learn More

"A man? You? You dream, Bayu, you dream, but all the same, you have not dreamed enough because you haven't made the journey to Mount Meru..."

The sharp words of his fiancée to the young African Muslim, Bayu, push him forward. He becomes impatient to achieve what generations of his ancestors, a clan of sculptors and artists, have done: To make the journey which all men of the clan must make, and to dream three dreams on Mount Meru, before he can take a wife. From a village near Mombassa, in Kenya, he must cross the forests to Mount Meru, in Tanzania, a huge distance across hostile terrain, made harder by his inexperience and innocence. Yet, step by step, the young sculptor throws himself into the adventure, which illuminates a wild universe--the strange beauty of the trees, the ever-present animals who alternately threaten and protect him on this increasingly magical but always dangerous voyage. And throughout, the eyes of the leopard--a figure which Bayu's ancestor held captivated and captive--protect him. Bayu's journey through the forests and mountains entrances us as we are absorbed into his world.François Devenne enchants with this remarkable tale. Eloquently translated by Lauren Yoder.Click Here to Learn More

The Nature of Christianity in Northern Tanzania explores the relationship between the region?s environment and social change during the pivotal, often over-looked German colonial period (1890-1916). The work connects changes in the landscape order and biogeography closely with the beginning Christianization of the three groups on the mountains ? the Chagga on Mt Kilimanjaro and the Meru and Arusha peoples of Mt Meru.

The work tells a story which is ordered, green and Christian. It looks at both new ideas and plants brought by the Germans to their colony in East Africa. The introduced German-like order and the exotic plants changed the landscape during the short period of German rule. However, the changes taking root in the African societies, driven primarily by the introduction of Christianity, led to an acceptance and adaptation of these imports. Religious change is one of the most profound elements of social change and it deeply impacted the world view of the Chagga, Meru and Arusha peoples. Within all three groups, their worldview was closely tied to religion ? there is no difference between the natural and social spheres nor the religious and secular worlds.

In the interaction between the German and Africans, the ideas, use of plants and even Christianity became altered, Africanized, and finally propagated by the African groups, helping to create the new African/European landscape. This heritage lives on up till today, growing on the landscape, nurtured by the changes in the societies of the Chagga, Meru and Arusha peoples on Mt Kilimanjaro and Mt Meru.Click Here to Learn More